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Can anyone help with advice as to teach my son to show his working in math problem solving. He insists that sometimes he can't, as "I don't know how I solved it". His answer is normally correct. He says that he knows how to "show working" and does when he can. His teacher emailed me detailing that he had given my son detention last week for "refusing" to show his working. He said her suspects he is using "alternative methods" but this will cause him to fail math at school. I know that as maths steps up in difficulty he will have to "show his working", so I would like to assist him if possible, but he is insistent that he "CANNOT" as he does not know how he solved it. Any ideas? He is 13.

I am at my wits end, and more detentions have been threatened.


Stephanie

Maybe your son will starting showing his work on his own when given math problems that require multiple steps and which he cannot do in his head. I would tell the teacher she should not be assigning detention for your son's "offence" and talk to the principal if necessary. OTOH, she would be within her rights in giving a student lower marks for not showing the work.

Homeschooling and private schooling are not cheap, but if a smart boy is coming to think of his teachers as adversaries, everything should be considered.

Your son should not lose confidence in his math ability or become turned off. While dealing with the school situation, I suggest supplementing his math education with an online program such as EPGY or Alcumus http://www.artofproblemsolving.com/Alcumus/Introduction.php or Thinkwell . Computerized math programs advance students to new topics when they get the right answers, so "showing the work" will not be an issue.

Stephanie,
Our son is 12 and refuses to show his working for math also. What is starting to work a little is a discussion we had a week ago.

We started by telling DS that while he understood what he had done on his homework, we couldn't understand it. We talked about how when he answers a question in Other classes, he often has to write a sentence or two. Those sentences have to follow the grammar, spelling, and punctuation rules or they are not easily understood by anyone reading the answers. The same thing is true in math, if he doesn't write the answers out in a recognizable form and show his thought process then it is much harder for anyone reading his answers to understand what he did.

In DS's case, he can do so many of the steps in his head that it looks like he picked an answer out of the air. So we are having DS teach us the material and we only write down the steps he explains. Once he saw the answers we were getting following his directions, he started getting a little better about showing more of his work. However, it is still a work in progress

Maybe it's time to introduce the concept of doing a proof. I know proofs are out of fashion in high school geometry these days, but they are very important past calculus. Math is all about showing how you got the answer.

Of course, if he's just doing simple addition, he can't really do proofs for those (until he takes number theory...)

In terms of alternative methods, if the point of the exercise is to show how different methods produce the same answer (thus getting a much deeper understanding of the number system), then it is important to do the problems different ways. If the point is just to get the answer, then it shouldn't matter how he solves it so long as the solutions are consistent.

... but at the same time, if someone is using an alternative method of problem-solving, it is all the more important to document it thoroughly.

That is a lesson in good communication skills, if nothing else; it's just not polite to do otherwise.


I like the approach mentioned above of having the child offer "instruction" in his/her method. That reinforces the idea that math and science problem solving can (and should be?) a collaborative exercise as often as it is a solitary one, and that being able to communicate well with others about what you are doing and why is essential to that collaboration.

That can be a hard lesson to learn with other age-mates, but I think it can work well with parental coaching. Great tip!

I understand your points, but some people (perhaps more boys than girls) with better math than verbal skills can solve certain math problems but not necessarily explain how they solved them. It is important not to turn them off.

There are also people who can quite articulately come up with answers that turn out to be wrong. Some of them become politicians.

Oh, I don't think it necessarily needs to rely on VERBAL explanation (though I have noticed that this is really in vogue right now in K-8).


In fact, it's probably even better to be able to break apart a problem into individual, logical mathematical/computational steps that follow one another even without the use of words.

A lot of 'gestalt' mathy kids struggle with that, particularly if they themselves don't need to do the computations step-wise. But it is important. Our solution when we run up against that one is to increase the problem complexity until DD can no longer do the entire problem in her head.

What kinds of problems are these, can you give an example or two? One alternative would be first to offer him challenges along the lines of:

"Explain why [answer to ... is ...]."

such that he very clearly cannot do the question unless he can explain, because the question *is* to explain. If he genuinely can't answer that kind of question, it's interesting, and worrying. If he can, the next step is to tell him that when he can work out the answer to the actual question in his head, he should set himself a question of the above form and write the answer to that as his answer.

We've explained to our DS that in research mathematics, where you are writing about a problem nobody has ever solved before, there is almost no point in stating an answer: the content of the job is to explain why that is the answer clearly enough that your colleagues can understand. Even for professionals, it can be hard to decide exactly which steps need to be explained and which don't, given the particular expected audience, and it is never too soon to start practising this, but at the very least, you should explain enough that someone who knows as much as you did when you started thinking about the problem but who has not studied the problem can follow your argument easily. DS sometimes complains about writing because he dislikes writing, but seems to accept the necessity. I think they key thing is to get over that nobody is disbelieving that he can get the answer without writing working - rather, writing the working is the point.

This is the approach that I've used with both my daughter and my students. With my students (middle and high school), I usually give them the line that I need to be sure that they are not doing math that I don't know, or don't remember.

What sometimes helps kids who are getting use to showing all of their work is to allow diagram-ing or "backwards" work to be shown.

For example: solve 3x+10 = 28

answer: 6, because 3 goes into 18 6 times
(which can then be used to get them to answer the question where did the 18 come from?)
18 is 10 less than 28

then I repeat their steps in the conventional order of solving problems.

It takes a few times of doing this before the student eventually feels confident/comfortable enough to do this themselves. However, most of the time these kids can break down the process backwards more rapidly than forwards and it is actually really cool to see their order of operations.

Just warn the teacher before he turns in the first assignment with the work done out "backwards" since many middle and high school math teachers will freak out when presented with work done out this way. (I'd explain it as the first step to getting him to do math the teacher's way and ask for understanding as you work towards what is required by the class.)